Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #95×C113 ⋊ C2 |
| Sylow | C113 ⋊ #385 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D113 |
| commutator | C22×C113 | C22×C4 |
| Fitting | #95×C113 | C2 |
| socle | C23×C113 | C23 |
| hypercenter | #95 | D113 |
| hypocenter | C113 | #385 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C23 ◁ #95 | |
234 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,113|a^2=f,b^2=f,b^a=bd,c^a=ce,g^a=g^112,g^c=g^1121148 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #385×D113, C2×#20298472, #28393012, … |
| maximal normal subgroup of | #385×D113, C2×#20298472, #28393012, … |
| maximal quotient of | #28388795, #28388808, #28388809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 423);